30 Questions
What is the relationship between the three statements r, s, and t in the proof of the theorem?
They are collectively exhaustive
What is the value of CD in the case where ∠C is obtuse?
b cos(180° - C)
What is the expression for a in the case where ∠C is a right angle?
c cos B
What is the implication of case (i) on the proof of the theorem?
r ⇒ q is proved
What is the value of BD in the case where ∠C is acute?
c cos B
What is the conclusion that can be drawn from the three cases proved in the theorem?
a = b cos C + c cos B
What is the primary purpose of the straight-forward approach in mathematics?
To establish a connection between a given statement and a conclusion
In the given example, what is the reason behind replacing x2 – 5x + 6 with (x – 3)(x – 2)?
To replace an expression with an equivalent expression
What is the underlying principle in the step 'x – 3 + 3 = 0 + 3 or x – 2 + 2 = 0 + 2'?
The property of adding equal quantities on either side of the equation
What is the conclusion that can be deduced from the statement 'x2 – 5x + 6 = 0'?
x = 3 or x = 2
What is the role of axioms in the straight-forward approach?
To help in deducing the conclusion from the given statement
What is the advantage of using the straight-forward approach in mathematical proofs?
It provides a logical and systematic approach to mathematical proofs
What is the primary method used to prove the first statement in the text?
Factorization
What is the definition of a one-one function?
A function that maps one element to one element
What is the purpose of the axiom in mathematical induction?
To prove that a proposition is true for all natural numbers
What is the symbolic equivalent of the argumentation process?
p ⇒ q
What is the primary step in proving that the function f(x) = 2x + 5 is one-one?
Assuming that f(x1) = f(x2) and then proving that x1 = x2
What is the result of dividing both sides of the equation 2x1 + 5 = 2x2 + 5 by 2?
x1 = x2
What is the symbolic form of the statement 'if a matrix A is invertible, then A is non-singular'?
p ⇒ q
If a matrix A is singular, what is the value of |A|?
0
What is the contrapositive statement of 'if a matrix A is invertible, then A is non-singular'?
If A is singular, then A is not invertible
What is the purpose of a counter example in proof by a counter example?
To disprove the statement
What is the mathematical notation used to represent the statement 'For each n, 2^(2n) + 1 is a prime'?
∀n ∈ N, 2^(2n) + 1 is a prime
What is the significance of the statement '~ (p ⇒ q)' in the context of proof by a counter example?
It is a disproof of the statement p ⇒ q
What is the conclusion of the proof that the set of all prime numbers is infinite?
The set of all prime numbers is infinite
What is the method of proof used in the first proof?
Proof by contradiction
What is the statement proved in Example 6?
The function f(x) = 2x + 5 is one-one
What is the contrapositive of the statement 'if f(x1) = f(x2), then x1 = x2'?
if x1 ≠ x2, then f(x1) ≠ f(x2)
What is the purpose of the proof by contrapositive?
To prove a statement by proving its equivalent contrapositive
What is the advantage of using proof by contrapositive?
It provides an alternative method to prove a statement
Understand the concept of a straight-forward approach in mathematics, where a chain of logical arguments leads to the desired proof. This quiz example shows how to prove that if x2 – 5x + 6 = 0, then x = 3 or x = 2 using algebraic methods.
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